Doubling power and proportion

Use in the book

The paragraph follows number, plane geometry, and the study of solids. The Athenian finds something divine and wonderful in the way power and its contrary turn about the double, so that the whole of nature stamps out form and kind in each proportion (990e). This is a compressed programmatic claim, not a worked demonstration that every natural object has already been mathematically derived.

The next sentences distinguish the double “in number,” one in relation to two in ratio, a double associated with power, and a development into the solid and tangible that passes from one to eight (991a). The edition's note interprets the sequence through dimensional increase: 1:2 in numbers, 1:4 in squares, and 1:8 in cubes. If a side is doubled, its square has four times the area and its cube eight times the volume. This gives an intelligible mathematical setting to the transitions without pretending that the Greek explains every intermediate operation in those terms.

The distinction between doubling a side, doubling an area, and doubling a volume must be maintained. A square twice another square's area has a side in the ratio square-root-of-two to one; it does not have a side twice as long. A cube twice another cube's volume has a side in the ratio cube-root-of-two to one, not two to one. These explanatory examples make the difference between dimensional scaling and making unlike quantities alike clear. They do not establish that the passage expressly poses a particular classical construction problem.

The “opposite” power is not independently defined in this brief passage. It can be discussed as the reverse or complementary operation in the proportional setting, but it should not be presented as an unambiguous technical name for a modern algebraic operation. Likewise, the language of form and kind here emerges from mathematical relationships; it is not by itself a proof of a separately existing realm of Forms.

The proportional sequence leads into the two means between six and twelve. Its educational role is to prepare recognition of one mathematical order across arithmetic, plane and solid magnitude, music, and celestial movement. Its obscurity is consequential: the lexicon can exhibit the edition's interpretation and its arithmetic, while preserving what the text leaves compressed.

Places to read

1. 990eπερὶ τὸ διπλάσιον — The double is the focus of the passage
2. 990eτῆς δυνάμεως καὶ τῆς ἐξ ἐναντίας ταύτῃ — Power and what stands contrary to it
3. 990eκαθ’ ἑκάστην ἀναλογίαν — The shaping proceeds in each proportion
4. 990e–991aεἶδος καὶ γένος ἀποτυποῦται πᾶσα ἡ φύσις — Nature stamps out form and kind
5. 991aἓν πρὸς δύο κατὰ λόγον — One to two as a ratio
6. 991aἡ κατὰ δύναμιν οὖσα — The next relation is described through power
7. 991aἀφ’ ἑνὸς εἰς ὀκτὼ διαπορευθεῖσα — The solid development passes from one to eight

Translation in this edition

“Power,” “proportion,” and “ratio” distinguish dynamis, analogia, and the mathematical logos. The 1:2, 1:4, 1:8 explanation belongs to the edition's note and is explicitly interpretive. The accepted reading text contains “one,” “two,” and “eight”; an explanatory diagram must not be mistaken for a fuller numerical table printed by the Athenian.

Mathematical one-to-two relation — λόγος (logos)

λόγος (logos)

Mathematical one-to-two relation. One in relation to two is stated according to ratio; this specialized sense differs from account and reason elsewhere. Attested: λόγον (991a). Doubling power and proportion.

Mathematical counterpart — ἐναντίος (enantios)

ἐναντίος (enantios)

Mathematical counterpart. The power contrary to the power around doubling is mentioned but not separately defined as a named modern operation. Attested: ἐναντίας (990e). Doubling power and proportion.

Proportional organizing relation — διπλάσιος (diplasios)

διπλάσιος (diplasios)

Proportional organizing relation. The double anchors the transition from number to square and cube relations and then the means; the text does not equate doubling length with doubling area. Attested: διπλάσιον (990e). Doubling power and proportion.

Mathematical power — δύναμις (dynamis)

δύναμις (dynamis)

Mathematical power. The power around the double, interpreted by the edition in the dimensional sequence involving squares; this specialized setting must not be imposed on all occurrences. Attested: δυνάμεως (990e). Doubling power and proportion.

Structured mathematical relation — ἀναλογία (analogia)

ἀναλογία (analogia)

Structured mathematical relation. Each proportion is a setting in which nature is said to stamp out form and kind, a programmatic claim rather than a completed derivation of every natural thing. Attested: ἀναλογίαν (990e). Doubling power and proportion.

Discerning proportion — ἐγκαθορᾶν (enkathoran)

ἐγκαθορᾶν (enkathoran)

Discerning proportion. Those who see into and think about the doubling relations can recognize their divine and wonderful character. Attested: ἐγκαθορῶσί (990e). Doubling power and proportion.

Nature's mathematical articulation — ἀποτυποῦσθαι (apotypousthai)

ἀποτυποῦσθαι (apotypousthai)

Nature's mathematical articulation. The middle verb figuratively gives nature the action of stamping form and kind in each proportion. Attested: ἀποτυποῦται (990e). Doubling power and proportion.

One to two in ratio (contextual phrase)

ἓν πρὸς δύο κατὰ λόγον (991a)

Logos here specifies a mathematical ratio. The relation 1:2 begins the interpreted progression through squares and cubes; it should not be conflated with doubling an area by doubling its side, which would instead quadruple the area. Doubling power and proportion.